Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T05:24:58.792Z Has data issue: false hasContentIssue false
  • English
  • Français

Precise Large Deviations for Sums of Random Variables with Consistently Varying Tails in Multi-Risk Models

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Shijie Wang  and
Wensheng Wang
Shijie Wang*
Affiliation:
East China Normal University and Anhui University
Wensheng Wang*
Affiliation:
East China Normal University
*
∗Postal address: Department of Statistics, East China Normal University, Shanghai 200062, P. R. China.
∗Postal address: Department of Statistics, East China Normal University, Shanghai 200062, P. R. China.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Assume that there are k types of insurance contracts in an insurance company. The ith related claims are denoted by {Xij, j ≥ 1}, i = 1,…,k. In this paper we investigate large deviations for both partial sums S(k; n1,…,nk) = ∑i=1k ∑j=1niXij and random sums S(k; t) = ∑i=1k ∑j=1Ni (t)Xij, where Ni(t), i = 1,…,k, are counting processes for the claim number. The obtained results extend some related classical results.

Keywords

Large deviationloss processconsistently varying tailsums of random variables

MSC classification

Secondary: 60F10: Large deviations 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 889 - 900
Copyright
Copyright © Applied Probability Trust 2007 

References

Berman, S. M. (1982). Sojourns and extremes of a diffusion process on a fixed interval. Adv. Appl. Prob. 14, 811–832.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Cai, J. and Tang, Q. H. (2004). On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Prob. 41, 117–130.CrossRefGoogle Scholar
Cline, D. B. H. (1994). intermediate regular and Π variation. Proc. Lond. Math. Soc. 68, 594–616.CrossRefGoogle Scholar
Cline, D. B. H. and Hsing, T. (1991). Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Preprint, Texas A&M University.Google Scholar
Denuit, M., Lefèvre, C. and Utev, S. (2002). Measuring the impact of dependence between claims occurrences. Insurance Math. Econom. 30, 1–19.CrossRefGoogle Scholar
Heyde, C. C. (1967). A contribution to the theory of large deviations for sums of independent random variables. Z. Wahrscheinlichkeitsth. 7, 303–308.CrossRefGoogle Scholar
Klüppelberg, C. and Mikosch, T. (1997). Large deviations of heavy-tailed random sums with applications in insurance and finance. J. Appl. Prob. 34, 293–308.CrossRefGoogle Scholar
Nagaev, A. V. (1969). Integral limit theorems for large deviations when Cramer's condition is not fulfilled. I. Theory Prob. Appl. 14, 51–64.CrossRefGoogle Scholar
Nagaev, S. V. (1973). Large deviations for sums of independent random variables. In Trans. Sixth Prague Conf. Inf. Theory Statist. Decision Functions Random Process, Academia, Prague. pp. 657–674.Google Scholar
Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Prob. 7, 745–789.CrossRefGoogle Scholar
Ng, K. W., Tang, Q. H., Yan, J. A. and Yang, H. L. (2003). Precise large deviations for the prospective loss process. J. Appl. Prob. 40, 391–400.CrossRefGoogle Scholar
Ng., K. W., Tang, Q. H., Yan, J. A. and Yang, H. L. (2004). Precise large deviations for sums of random variables with consistently varying tails. J. Appl. Prob. 41, 93–107.CrossRefGoogle Scholar
Tang, Q. H. and Yan, J. A. (2002). A sharp inequality for the tail probabilities of sums of i.i.d. r.v.'s with dominatedly varying tails. Sci. China Ser. A. 45, 1006–1011.CrossRefGoogle Scholar
Tang, Q. H., Su, C., Jiang, T. and Zhang, J. S. (2001). Large deviations for heavy-tailed random sums in compound renewal model. Statist. Prob. Lett. 52, 91–100.CrossRefGoogle Scholar